Assume N spin-1/2 particles, and only focus on the spin states. The dimension of the Hilbert space is then $2^N$. The ground state could be found by diagonalizng the Hamiltonian $H$. As is often stated, it is hard to find the ground state of $H$ when $N$ is large. My question is what is the largest $N$ that could be diagonalized (or has been diagonalized) on super computers? Assume $H$ is Hermitian and diagonalizable, but otherwise a "random" matrix with no special properties. I believe this $N$ should be importatnt for such claims, but has not found a satisfactory answer through a quick search of Google, or chatting with ChatGPT...
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If H were a completely random matrix, it wouldn't even be possible to store it (or rather, storing it would be the actual bottleneck). This is not the relevant scenario. – Norbert Schuch Sep 11 '23 at 09:35